This work provides an explicit relativistic non-linear differential equation to estimate the ground-state electron density, and especially its directionality dependence, for large clusters of heavy atoms, such as Pb, at their experimentally measured equilibrium geometry. The study embodies the early theory of Vallarta and Rosen, which seems to us to build a firm foundation on relativistic semi-classical many-electron theory. Assuming a finite nuclear radius for the heavy atoms would be advisable in subsequent numerical applications. (C) 2014 Published by Elsevier B.V.